Mock AIME II 2012 Problems/Problem 4
Let be a triangle, and let , , and be the points where the angle bisectors of , , and , respectfully, intersect the sides opposite them. Given that , , and , then the ratio can be written in the form where and are positive relatively prime integers. Find .
Let and . We use the Angle Bisector Theorem twice to get two different equations relating and . From the angle bisector , we have the proportion , or . From the angle bisector , we have the proportion , or . Multiply the second equation by to get , then plug this expression for into the first equation to get . Solving this equation gives us . Finally, the ratio is, by the Angle Bisector Theorem, equivalent to the ratio , which is equal to so the answer is .